When I first learnt about e, I just treated it as another number (as far as I know, it doesn't even have a natural definition), but how is it that exp(x) is so important and keeps showing up at various places in mathematics?
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If you differentiate exp(x), it will remain the same. – Hwang Apr 17 '17 at 15:21
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$e$ is very important because it keeps showing up in various places in mathematics. Specifically, calculus though.
Calculus is the study of the rate of change, and the first time you have probably learned about the number $e$, you see how much it plays a key role in the rate of change of bank interest, or exponential growth/decay. Since calculus is the study of rate of range, $e$ is bound to appear in many aspects across calculus, whether that be derivatives, integration, polynomials, many advanced topics too like differential equations.
There is many cool properties of $e$ too.
K Split X
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One beautiful occurrence I just found out today:
Consider the equation: $$x^y = y^x $$ $${x,y \in R^+}$$
For every positive value of x, there are two values of y that satisfy the above equation ONLY after x= e and vice versa.
To admire this amazing result, look at the graph of the equation on Desmos.
Sid
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To be clear, $y=x$ is a solution for any fixed $x\in\mathbf{R}^+$. When $x>1$, $x\ne e$ there is a second $y$ that solves the equation. This follows by writing the equation as $\frac{\ln x}{x}=\frac{\ln y}{y}$ and noting that $\frac{\ln x}{x}$ is strictly increasing for $x\in(0,e)$ and becomes positive for $x>1$, while it is strictly decreasing and approaches $0$ for $x\in(e,\infty)$. – Will Orrick Mar 29 '23 at 19:57